Consider Two Independent Continuous Random Variables X and Y With Densities Find X Y
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7. Consider two continuous random variables X and Y with the following joint probability density function f(x,y) =axy,0< x <2,0< y <4. a. Calculate the value of a. b. Obtain the joint cdf of X and Y. c. Are X and Y independent?
Related Question
The joint probability density function of $X$ and $Y$ is given by $$ f(x, y)=\frac{6}{7}\left(x^{2}+\frac{x y}{2}\right) \quad 0<x<1,0<y<2 $$ (a) Verify that this is indeed a joint density function. (b) Compute the density function of $X$. (c) Find $P\{X>Y\}$. Chapter 6 Jointly Distributed Random Variables (d) Find $P\left\{Y>\frac{1}{2} \mid X<\frac{1}{2}\right\}$. (e) Find $E[X]$.
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Video Transcript
Hi guys is this problem you're given that half of X and y is equal to 6/7 times X squared plus X. Y over two. This is for ex Maura's NGO and less than one. And Why? More than zero and less than two. Okay, so now um we have integration over X from 0 to 1 is an integration from 0 to 2 for f of X dy dx. So this is X over seven X squared plus X. Y over two. Do I? The X. This is equal to one. Okay, so this is 6/7 times integration from 0 to 1. Four X squared Y plus X over two times y squared over two. And then we have the limits to Anzio and then we integrate relative to X. Okay, so this is one. So this is 6/7 times um two x cubed over three plus X squared over two and suddenly have the limits one and zero. This is equal to one. So this is text over seven times 7/6. This is true. Since the true function can be written as double integration over X and y is equal to one. So now we need to find the aftereffects. This is integration over Y from 0 to 24. 6/7 times X squared plus X. Y over two dy So this is 6/7 Times two x squared plus X. Okay. Okay so probability of X more than Y. This is integration from X 20 2. 1. Then integration over life from 0 to X for 6/7 times x squared plus xy over to dy dx. So in party we need to find um this probability. Okay, so here it's X axis and here the Y axis. So when y is equal to tool And X is equal to one. Okay. So here we have a rectangle That's high off to and rights of one. So um mhm. This is all .5 and this is one. Okay. So. Okay. Okay, mm hmm. This is where X less than y. Okay. So this probability is just um 6/7. It's an integration from 0 to 14 X X squared Y plus xy squared over four. Okay then we have the limits X and zero within the X. Okay, so this is sex over seven times integration from 0-145 XQ Pull that four. And the x. So this is 30/20 eight times Explore 4/4. And we have that and it's one and 0. So this probability is 4.2678. Okay. And party we have a plot for X and Y. Okay, so mhm This is X is equal to one Here is 0.5. He is 4.5. One and two. Okay. Okay. And we need to find the probability of this part. Okay, so uh the shaded part here is the probability of X. Less than 0.1 over to intersect. Why more than 1/2. Okay, so this is the probability of Integration of from 0 to 4.5. Then integration from 0 to 2 for F. of X. And why do I the x minus? Ah Integration from 0 to 4.5 is an integration from 0 to 4.5 for F. Of X and Y dy dx. Okay. So um to find the probability of um Y not X. Why more than one over to given that X less than 1/2. So its probability of intersection between X and y Over the probability of X less than 1/2. Okay so now we need to find F. X. Of X. So this is integration from 0 to 2 4, 6/7 times X squared plus X. Y over two dy so this is 6/7 times two X squared plus X. Okay sen Now we need to find integration from 0 to 4.5. It's an integration from 0 to 2 for um I have X and Y. Okay or simply This is 6/7 times X squared plus. Xy over to D Y. Dx. So this is equal to integration from 0 to 4.5 four. Um 6/7 for two x squared plus. Um X. Dx. Okay so this is equal to 1/7 times 4/8 plus. 6/8. So this is 10/56 Which is approximately 0.17 86. Okay now we need to consider the probability of From 0 to 4.5 and then from 0 to 4.5 for f of X. The X. Which is 6/7 times um times what times X squared plus X. Y over two. Then we have dy dx. Okay, So this is equal to 6/7 Times 1/48 plus 1/1- eight. So this is equal to all points. 08455. Okay. Now we can find the probability of why More than 1/2 given that X less than 1/2. So this is 4.1786 -0.02455 Over 1/7 tons integration From 0 to 4.5 for 12 x squared plus six X D X. Okay, So this probability is all .8625 but then we need to find someone of X. So the mean of X is integration from 0 to 1 for X times F of X. Mhm Which is um 12 X squared plus six X over seven in the X. So this is 1/7 times three x power four plus 2 x killed. And then we have some limits on Anzio. Okay. It is then we have that this is equal to 1/7 times three plus two. So this is 5/7. And that's it
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